Local profiles and multiscale elliptic problems with defects.

Summary We present a possible approach to approximate at both the coarse and fine scales the solution to an elliptic equation with oscillatory coefficient when this coefficient consists of a “nice”, say periodic, function that is locally perturbed. The approach is based on a local profile, solution to an equation similar to the corrector equation in classical homogenization. The well-posedness of that equation is explored, in various functional settings depending upon the locality of the perturbation. Some related problems are discussed.